Problem 11 · 2019 AMC 8
Medium
Counting & Probability
complementary-counting
The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
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Answer: D — 39 students.
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Hint 1 of 2
Add the two class sizes: 70 + 54 = 124, but there are only 93 students. The extra 124 − 93 = 31 isn't a mistake — it's the kids counted twice because they're in both classes.
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Hint 2 of 2
Once you know how many take both, "math only" is just the math total minus the both-takers (subtract off the shared slice of the Venn diagram).
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Approach: the overcount equals the overlap
- Every student is in at least one class, so the two lists together should cover all 93 people — yet 70 + 54 = 124. The 124 − 93 = 31 surplus is exactly the students counted in both lists, so Both = 31.
- "Math only" peels off the shared part: 70 − 31 = 39.
- Why this transfers: whenever two groups together exceed the whole, the excess is the overlap — that's inclusion-exclusion (|M| + |F| − |both| = total), and a two-circle Venn diagram makes the "only" regions obvious.
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